This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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Lists of sets as objects of ZF axiomatics

I have a naive question about foundations of mathematics. A common opinion of most mathematicians is that the essential part of mathematics can be reduced to ZF(C) axioms. I do not quite understand ...
26
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6answers
1k views

Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?

I understand that naive set theory, whose axioms are extensionality and unrestricted comprehension, is inconsistent, due to paradoxes like Russell, Curry, Cantor, and Burali-Forti. But these all ...
3
votes
1answer
59 views

Categorical Foundations text

I've heard that someone's thought up a way of using category theory, involving something called topoi, as a foundation for mathematics. If this is true then are there any texts which explain such a ...
5
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2answers
79 views

When can variables simply be variables?

This may seem a somewhat strange question, but I've been tying myself in knots about it recently. When constructing a polynomial ring, you must formally define a polynomial as an ordered ω-tuple, ...
3
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2answers
263 views

Why is it important, that mathematics can be formalized in set theory?

Why is it important, that mathematics can be formalized in set theory? As one can read in the thread Are there areas of mathematics that cannot be formalized in set theory? Today known mathematical ...
9
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3answers
245 views

Why is something not a field if it's a proper class?

Why is it a convention to say that, for example, the nimbers and surreal numbers aren't fields because they don't form sets? Is this just pedantry?
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2answers
52 views

The class of all functions between classes (NBG)

Is it possible in NBG (von Neumann-Bernays-Gödel set theory) to construct the class of all functions $X \to Y$ between two (proper) classes $X,Y$? I guess that this does not work. In the special case ...
0
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2answers
57 views

Construction of Natural Numbers

I am trying to prove that the natural numbers can be constructed from the product of a power of $2$ and an odd number. For all $n \neq 0$ in the natural numbers, $n = (2k+1)(2^p)$, where $k$ and $p$ ...
3
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1answer
119 views

Definiton of Limit and Foundational problems.

I am new to Category theory and I have a quite strong foundational problem. For example, let's start from the definition of Limit stated by wikipedia ...
2
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1answer
128 views

I need help understanding Frege's definition of number

I have really been trying to understand Frege's definition of a number or at least gain a strong intuition of it. However, my attempts have not been fruitful. If someone could help me it would be much ...
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1answer
52 views

Defining integer sum without using infinite sets

In ZFC minus infinity (let us call this system $T$), one can still define ordinals, and then define integers as ordinals all of whose members are zero or successor ordinals. Combining the power set ...
4
votes
1answer
89 views

Integer induction without infinity

In ZFC minus infinity (let us call this T), one can still define ordinals, and then define integers as ordinals all of whose members are zero or successor ordinals. I am looking for a formula $\psi$ ...
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2answers
34 views

how can we express finiteness as a first order property?

I don't know much about set theory but I read that in ZFC a set is finite when there are no bijections from the set to a proper subset of itself. It seems to me however that quantifying over subsets ...
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2answers
58 views

Function, Relation, Operation and Cartesian Product

An operation is a kind of function. A function is a kind of relation. A relation is a subset of a Cartesian product. A Cartesian product is an operation. Back to 1. It seems to me that there's ...
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1answer
112 views

Why does 2+2 equal to 4? [duplicate]

The question is in the title. I am very appreciative of any time and concern put into belaboring this relatively little problem.
55
votes
9answers
3k views

Why is the construction of the real numbers important?

There are a lot of books, specially in Real Analysis and set theory, which define the real numbers by Cauchy sequences or Dedekind cuts. So my question is why don't we simply define the Real numbers ...
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votes
1answer
34 views

Can boolean logic compute any sort of mathematical operation?

Computers fundamentally do logical operations on the input and memory they have (as far as I know). Computers are used by mathematicians to do all sorts of mathsy operations (as far as I know). Does ...
3
votes
0answers
126 views

How should one understand the foundation of set theory?

I have read the answer of Carl Mummert for the question on how to avoid circularity. I would like to ask further as I want to study models of set theory. As I understand, with say assuming the ...
3
votes
1answer
120 views

Does internalization loses informations everywhere?

It is well known that a group object in Grp is necessarily abelian. This can be understood as "internalization loses information". Indeed, if one was to study group theory by looking at group objects ...
2
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1answer
55 views

A nonempty class of isomorphic groups defines a group

The context of this question is from the definition of the sporadic Mathieu group $M_{23}$, which (in one possible definition) is the stabilizer of a point in $M_{24}$, which is a certain subgroup of ...
3
votes
2answers
225 views

What is so special about the real and complex numbers?

When I was studying linear algebra, the first thing we were introduced was the idea of fields. In studying analysis (and when studying inner product spaces etc), we restricted our possible fields to ...
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3answers
59 views

Abstracting objects

I understand that one way of defining a mathematical object such as a group is to take an object we already know to exist, for example the integers, and take away some properties from them. This is ...
0
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1answer
104 views

Why not allow creativity of definitions?

It appears to me that a fair number of issues with allowing ZFC to work with other mathematical topics is that one cannot phrase certain definitions inside ZFC. Would not this be fixed by allowing ...
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0answers
21 views

Resources for studying different set theories [duplicate]

Ok so lately I have been fascinated about general structure of mathematics and I have read some books on set theory. I have gone through the Endertons introductory book on set theory which operates ...
4
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1answer
62 views

Does $\lambda^2 \leq \kappa^2 \Rightarrow \lambda \leq \kappa$ imply the axiom of choice?

I'm aware that the statement "for all cardinals $\kappa$, $\kappa^2 = \kappa$" is equivalent to the axiom of choice (I believe this was proved by Tarski). More generally, does anyone know if the ...
3
votes
2answers
241 views

stratification (typage) of logic and syntax at the same time: is such a dream feasible? [closed]

This post is more philosophical than formal, yet I think it's an important question. There's an idea I have for long times already that would consist, in some sense, in doing a "theory of theories". ...
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2answers
233 views

Groupoids more fundamental than categories, really?

I've skimmed through a survey by Thierry Coquand on univalent foundations. It is claimed that "groupoids are more fundamental than categories". And that categories can be seen as groupoids equipped ...
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1answer
74 views

Explanation of $\mathrm{ZFC} + \lnot \mathrm{Con(ZFC)}$

I read in JDH's answer to this mathoverflow question that $\mathrm{ZFC}$ is equiconsistent with $\mathrm{ZFC} + \lnot \mathrm{Con(ZFC)}$. But this second statement is a bit weird. What does ...
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1answer
87 views

Is it possible to construct ZFC set theory inside category theory?

It's entirely possible I don't understand what I am talking about, but I know that ZFC stands as a good foundation for much of mathematics and that category theory stands as a good foundation for ...
3
votes
1answer
155 views

Which areas of modern mathematics don't use ZFC set theory?

I've heard that Algebraic Geometry requires something called Category Theory, which itself requires an extension of ZFC called Tarski-Grothendieck set theory, and that got me wondering. Which areas ...
6
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0answers
47 views

What is the actual significance of the lambda calculus for the formalization of math?

The Simply Typed Lambda Calculus was proposed initially as a foundational system for the formalization of mathematics. As such, I would expect that soon there would be attempts to implement most of ...
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0answers
33 views

Categorical presentation of “the theory of structure in Set”

I have been thinking about colimits in finitely accessible categories. Here is a paper that abstracts the notions of Domain theory up to categories themselves. This means that we have notions of ...
5
votes
1answer
160 views

Is category theory constructive?

Roughly: This question concerns the process and the constructive nature of formalizing and proving category theoretic statements within $\textsf{ZFC}$. $\textsf{ZFC}$ can only talk about sets, ...
4
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0answers
45 views

Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
4
votes
1answer
73 views

At the lowest level, can mathematical proofs be reduced to substitutions and rewrites?

This is the impression I got from reading Douglas Hofstadter's book, Godel, Escher, Bach. He spoke of being able to design a proof checker that applied axioms and theorems to determine if a statement ...
3
votes
1answer
165 views

Good book on foundations - axiomatic set theory

I'm currently planning on reading Suppes' Axiomatic Set Theory, because I'm interested in finding out what the currently accepted foundations of mathematic are. Is this a good book for doing so? What ...
2
votes
4answers
115 views

Counting numbers vs Natural numbers; Peano Axioms

I can feel that my question is going to be a somewhat lengthy one, but I will try my best to deliver it in as short a form as I can manage. So to begin, I've always thought that the numbers such as ...
0
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1answer
50 views

Immediate consequence in Gödels incompleteness paper

In the famous paper, “On Formally Undecidable Propositions of PM”, $c$ is defined as the immediate consequence of $a$ and $b$ if $a$ is the formula $\lnot b \lor c$. How does this relate to the ...
2
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1answer
64 views

Seemingly circular definition of a collection

In the book I'm reading on set theory right now, I'm given this definition: "If $\Phi(x)$ is a formula then the term $\iota !y(\forall x)(x\in y\iff \Phi(x))$ is abbreviated $\{x:\Phi(x)\}$ and read ...
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2answers
79 views

Is it possible to map mathematics without advanced set theory?

I wish to make a large digraph (network) linking various proofs together in mathematics from, say, the definition of a group to Galois theory. I got in my head that I wanted to do this after reading ...
0
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1answer
125 views

Give a recursive definition of the operation of multiplication of natural numbers using the operations s and addition.

Give a recursive definition Basis: m,n subset of N(natural numbers) A contains N (0,m) subset of A If n = 0, 0*m = 0 (m + 1) * x = mx + x Recursive Steps:
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2answers
68 views

Set membership as a binary relation: self-reference in ZFC?

First of all, I want to make it clear that I am a real novice when it comes to Mathematical logic, so any answer that involves jargons of the discipline will most likely make no sense to me at all... ...
3
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0answers
63 views

Bachmann's construction of the real numbers

On page 44 of this book an approach to constructing the real numbers as equivalence classes of nested rational intervals is outlined and attributed to Bachmann. The outline in the book is very ...
0
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1answer
87 views

In Whitehead & Russell's PM, what makes anything true or false?

It seems that truth and falsehood are fundamental to meanings and types. A proposition is defined as anything that is true or that is false. PM defines truth as "consisting in the fact that there is ...
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0answers
67 views

Do all mathematical fields require an algebra?

My understanding is that "algebra" refers to a specific field in mathematics. Here is Wikipedia's introduction: Algebra is one of the broad parts of mathematics, together with number theory, ...
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2answers
167 views

Why don't we use Presburger's arithmetic instead of Peano's arithmetic?

I was reading about quantifier elimination and discovered the Presburger Arithmetic, the article mentions two points about it: It is decidable, complete and consistent. It omits multiplication ...
0
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1answer
70 views

Foundations of math and primitive terms leading to Russells paradox

If the statement "Is an element of" is a primitive term (is not defined/you cant determine its truth) then how do you determine the truth of statements such as "If $x$ is in $S$, then $x$ has property ...
5
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3answers
233 views

Intuitionistic Logic and Classical Logic on the proof of (A or B)

In intuitionist logic, a proof of (A or B) means a proof of A, or a proof of B, whereas in Classical logic, a proof of (A or B) may be done withouth either proving A or proving B. I'm trying to ...
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1answer
67 views

What does if-then has to do with not being true?

I'm reading Chihara's: Constructibility and Mathematical Existence. It says: An even more radical view rejects the assumption that mathematics is true—at least in the straightforward way that ...
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1answer
48 views

A question on Mortiz Pasch works in foundation of mathematics

I am reading "Introduction to the Foundation of Mathematics" ,by R L.Wilder(2nd Ed.), where Mortiz Pasch's works are described in paragraph 1.5. There is a quotation in that paragraph- " For if,on ...